3.9.2: Exercise S.9
- Page ID
- 148598
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Effect of Fertilizers on the Growth of Tomato Plants
- A statistics student who enjoys gardening conducted an experiment to examine the effect of using a new eggshell fertilizer on the growth of tomato plants. The student randomly assigned sixteen Black Cherry tomato plants to two conditions: (a) eight plants were fertilized with a traditional organic fertilizer, and (b) eight plants were fertilized with an organic fertilizer based on eggshells. The student measured the growth (in millimeters) of each plant each week. The tomato plants were randomly planted throughout a large plot of land. The tomato plants were the same size when planted and were planted on the same day. The same amounts of fertilizers were used each week over a 4-week period.
The table below displays the average growth rate of each plant in the two fertilizer groups (in millimeters per week).
| Traditional Fertilizer | Eggshell Fertilizer |
| 4.9 | 5.9 |
| 5.2 | 5.1 |
| 4.8 | 5.5 |
| 4.7 | 5.9 |
| 4.9 | 4.8 |
| 4.8 | 4.9 |
| 4.6 | 5.7 |
| 5.9 | 5.6 |
(a) The experiment was conducted on two samples. Define the samples and the corresponding populations of interest.
(b) What is the explanatory variable in this experiment? What two conditions were assigned to the explanatory variable?
(c) What is the response variable in this experiment?
(d) Experimental controls are implemented to eliminate confounding variables and ensure that changes to a response variable are influenced solely by changes to the explanatory variable. Identify an experimental control and state the corresponding confounding variable that is being addressed.
(e) Compute the sample mean and sample standard deviation for each group of tomato plants. Round the values to two decimal places. Compute the difference in sample means: mean of eggshell fertilizer sample minus mean of traditional organic fertilizer sample.
(f) The student performed a hypothesis test using the following hypotheses:
Null Hypothesis: Mean growth rate of plants with eggshell fertilizer = Mean growth rate of plants with traditional organic fertilizer.
Alternative Hypothesis: Mean growth rate of plants with eggshell fertilizer > Mean growth rate of plants with traditional organic fertilizer.
The null hypothesis states that the type of organic fertilizer used has no effect on tomato plant growth. The alternative hypothesis states that the eggshell fertilizer increases tomato plant growth compared to the traditional organic fertilizer. The hypothesis test returned a P-value of
0.026. Interpret this P-value.
(g) What can we decide about the null and alternative hypotheses?
(h) What can we decide about the two fertilizers?